The instantaneous velocity is the velocity of an object at a certain time. If given its position before, during, and after the required time, the instantaneous velocity can be estimated. While estimates of the instantaneous velocity can be found using positions and times, an exact calculation requires using the derivative function. The instantaneous velocity is not the same thing as the average velocity.

I want to talk about the conceptof instantaneous velocity.Let me step aside for a second and talk about the difference between instantaneousspeed and average speed.

Let's say you're on a road trip.You drive 300 miles and ittakes you five hours.Your average speed is 300 miles dividedby five hours, or 60 miles per hour.But as you're driving, you look at the speedometer.It's not telling you averagespeed it's telling you instantaneous speedand it's going to vary a lot.It may average out to be 60 but it varies as high as 70, as low as 0 if you stop.That's the difference between instantaneous speed and average speed.Now how do you calculate those things?Well, that's what we're goingto talk about now.

Let's go back to our pumpkin example.A pumpkin is catapulted into the air.Time T is in seconds.Height is in feet and here'sa small table of values.Now, suppose I wanted to find the instantaneousvelocity at T equals 0. Well,I could get a decent approximation bycoming up with the average velocityover this interval here.And it would be 200 minus118, which is 82.Divided by 1. So 82 feet per second.I can also use the average velocity fromtwo to three and that's 250 minus 250divided by 1. So 50 feet per second.

Now these approximations are usinga change in T of 1. Both of them.2 minus 1 is 1. Let's see what happensas we narrow this delta T value.So here I have a table that includesT equals 2. But now I have a pointin the left and a pointin the right.They're much closerto 2. 1.9.This is a delta T value of .1.

Now, what are the average velocities?On the left I get 67.6 feet per second, andon the right I get 64.4 feet per second.These values are getting alot closer to each other.Let me go a step further.Let me go to 1.99 and2.01.Here the delta T value is .01.And the average velocity on the left is66.16 and on the right 65.84 toto the nearest unitthese both round to 66.So you might say that the instantaneousvelocity is approximately 66 feet persecond.

And it turns out that this is exactlyhow we find instantaneous velocity.As delta T approaches 0, the length ofthe time increment that we're takingaverage velocity, as that increment goesto 0, average velocity approachesthe value of instantaneous velocityat that particular time.

And that's how we calculateinstantaneous velocity.It's always a limit of average velocitiesas delta T goes to 0.